On relating Eulerian and Lagrangian velocity statistics: single particles in homogeneous flows

Various theories seeking to relate the velocity statistics of Lagrangian particles to the statistics of the Eulerian flow in which they are embedded are examined. Mean particle drift, mean-square particle velocity and the frequency spectrum of velocity are examined for stationary, homogeneous and joint-normally distributed Eulerian fields. Predictions based on a third-order weak-interaction expansion, the successive approximation procedure of Phythian (1975), the quasi-normal approximation of Saffman (1969), the parametrized model of Saffman (1962), and a new procedure based on a statistical estimator of the kinematic equation are compared with simulations of particle motion in one-dimensional flow. Only the statistical estimator produces both acceptable mean-drift and frequency-spectrum predictions.